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NumberFieldFundamentalUnits[a]

gives a list of fundamental units for the field generated by the algebraic number a.

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NumberFieldFundamentalUnits

NumberFieldFundamentalUnits[a]

gives a list of fundamental units for the field generated by the algebraic number a.

Details

Examples

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Basic Examples  (1)

Find generators of the group of units for :

Wolfram Language code: NumberFieldFundamentalUnits[Sqrt[2] + Sqrt[3]]

Scope  (4)

Radical objects:

Wolfram Language code: NumberFieldFundamentalUnits[Sqrt[2]]

Root objects:

Wolfram Language code: NumberFieldFundamentalUnits[Root[-1 + 2#1 ^ 3&, 1]]

AlgebraicNumber objects:

Wolfram Language code: NumberFieldFundamentalUnits[AlgebraicNumber[Root[-3 + #1^3&, 1], {2, 1, -1}]]

NumberFieldFundamentalUnits automatically threads over lists:

Wolfram Language code: NumberFieldFundamentalUnits[{Sqrt[2], E ^ (I Pi / 4)}]

Applications  (1)

Table of fundamental units of for positive square-free :

Wolfram Language code: Table[{d, RootReduce[NumberFieldFundamentalUnits[Sqrt[d]]]}, {d, Select[Range[2, 13], SquareFreeQ]}]

Properties & Relations  (3)

Fundamental units are algebraic units:

Wolfram Language code: NumberFieldFundamentalUnits[Root[-1 + 2#1 ^ 5&, 1]]
Wolfram Language code: AlgebraicUnitQ /@ %

Use ToNumberField to find the fundamental units for :

Wolfram Language code: primitive = ToNumberField[{Sqrt[3], Sqrt[7]}, All][[1, 1]]
Wolfram Language code: NumberFieldFundamentalUnits[primitive]

The number of fundamental units can be obtained from NumberFieldSignature:

Wolfram Language code: primitive = ToNumberField[{Sqrt[3], Sqrt[7]}, All][[1, 1]]
Wolfram Language code: {s, t} = NumberFieldSignature[primitive]

The number of fundamental units:

Wolfram Language code: s + t - 1
Wolfram Language code: Length[NumberFieldFundamentalUnits[primitive]]

Neat Examples  (1)

Plot the fundamental units of for positive square-free integers :

Wolfram Language code: ListLogPlot[Table[{d, First@NumberFieldFundamentalUnits[Sqrt[d]]}, {d, Select[Range[2, 200], SquareFreeQ]}], Filling -> Axis]
Wolfram Research (2007), NumberFieldFundamentalUnits, Wolfram Language function, https://reference.wolfram.com/language/ref/NumberFieldFundamentalUnits.html.

Text

Wolfram Research (2007), NumberFieldFundamentalUnits, Wolfram Language function, https://reference.wolfram.com/language/ref/NumberFieldFundamentalUnits.html.

CMS

Wolfram Language. 2007. "NumberFieldFundamentalUnits." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/NumberFieldFundamentalUnits.html.

APA

Wolfram Language. (2007). NumberFieldFundamentalUnits. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/NumberFieldFundamentalUnits.html

BibTeX

@misc{reference.wolfram_2026_numberfieldfundamentalunits, author="Wolfram Research", title="{NumberFieldFundamentalUnits}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/NumberFieldFundamentalUnits.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_numberfieldfundamentalunits, organization={Wolfram Research}, title={NumberFieldFundamentalUnits}, year={2007}, url={https://reference.wolfram.com/language/ref/NumberFieldFundamentalUnits.html}, note=[Accessed: 12-June-2026]}